Total NERD Alert. Cos I CAN. AND I do and I always WILL

4.1 Directed Acyclic Graph

6.13 Directed-acyclic-graph (DAG)

This Direct Acyclic Graph (DAG) ledger transaction works on a peer-peer validation. Any node that wants to perform a transaction must verify at least one previous transaction with the new transactions (Cao et al., 2020). DAG is a distributed peer grading system. Instead of having a group of miners to validate the transaction, DAG uses the very transaction that the user performs to confirm each other's transaction. The advantage of using the DAG is it increases the operation speed and does not require miner selection. It is reliable and increases the security of the blockchain.

Though most of the Blockchain design is distributed and decentralized, centralization of the power and control can occur because it requires a witness node to mitigate the attacks to the blocks. To overcome these central points of control and failure some advanced consensus algorithms are required.

3.12 Directed Acyclic Graphs

Although Directed Acyclic Graphs or DAGs are basically a form of data structures and are not real blockchain networks, we decided to add them to this paper as they are widely being used in successful cryptocurrencies. Also knowing about the functions of DAG can help readers to better understand the blockchains. NXT, IOTA and IoT Chain are among the most successful applications of DAG. In real blockchain networks, transactions are stored in a chain of networks but in DAGs transactions are stored topologically in a graph. Fig. 3.a shows a cyclic and non-directed graph and Fig. 3.b presents an acyclic graph.

An acyclic graph is a graph that has no cycle. In an acyclic graph, information cannot be passed from one node to another node and return to the original node without encountering a node more than once. A Directed Acyclic Graph is an acyclic graph that the information can only pass through a pre-defined direction.

Due to their blockless structure, DAGs are considered as blockchains without blocks (Kotilevets et al., 2018). Cryptocurrency transactions are verified and added to the network in a way that is faster than PoW and PoS based networks, as there is no need to store them inside a real block and then verify the whole block (Benčić & Žarko, 2018). In a blockchain, an arbitrary period of time needs to be set to ensure the main chain remains viable. This waiting time is known as a block time and gives the network a time to consolidate and verify which branch of the chain is correct. However, in the DAGs, as long as the information is directed in the same way, nodes can exist in parallel.

This type of network would give us the ability to eliminate the need for block times and verify the transactions more quickly. The result is a fast, scalable and completely decentralized network (Kotilevets et al., 2018). Blockchains are susceptible to double spending, soft and even hard forks, however, in DAGs the validation of a certain transaction is decided by the number of the transactions behind it. This makes a DAG system faster and guarded against a double-spending attack.

In terms of the width of the network, adding the transaction to an earlier transaction every time would make the network too wide. On a DAG each validated transaction needs to link to an existing and new transaction of the network. When a transaction happens in a complete DAG network, the network would choose an existing later transaction to link to. This approach would keep the network width within a certain range that can support quick validation for transactions. IOTA has proposed its own algorithm named Tangle to control the width of the network (Bramas, 2018; Sarfraz, Alam, Zeadally, & Khan, 2019).

There is no mining process on a DAG network and therefore, there is no dependency on special hardware hence power consumption is very low. The validation of the transactions happens almost instantly. The transaction fee could be very low and fast. So, this makes a DAG network friendly to small and even microtransactions or payments. IoT chain, for example, can handle over 10,000 transactions per second. These characteristics of a DAG network along with its ability to defend against a 51% attack has made it a perfect approach for Internet-of-things and Machin-to-Machin communications (Bester, 2018).

A directed acyclic graph of YV is a graph of arrows in dV nodes without directed cycles, i.e., starting from any one node it is impossible to return to this node by following any path in the direction of the arrows. The (i, j) arrow is missing in it if

(2)i∐j∣parents ofi.

Nodes from which an arrow points directly to node i are called the parents of i.

For a Gaussian distribution, the independencies show as zero coefficients in recursive linear equations having independent residuals. More precisely, we have equations AYV=ε, with residuals ε having zero mean and covariance matrix, cov (ε), being a diagonal matrix T. Further, A is upper triangular with ones along the diagonal, giving Σ−1=ATT−1A. Therefore (A, T−1) is a triangular decomposition of the concentration matrix and off-diagonal elements in row i and A are negative values of regression coefficients when regressing Yi on all its potentially explanatory variables, i.e., variables ordered to have indices larger than i. Thus, the independencies show as zeros in A. If only a directed acyclic graph is given, typically more than one set of recursive equations is compatible with it, i.e., with several different orderings of the variables, an upper triangular matrix can result which reflects the same independence structure as the given graph.

However, often a unique full ordering is provided from substantive knowledge about how the data could have been generated. This was how the geneticist S. Wright introduced and used path diagrams for recursive linear relations. For other than linear relations, a generating process in terms of univariate conditional distributions still determines uniquely the edge matrix in a directed acyclic graph and a corresponding factorization of the joint density. Separation criteria for directed acyclic graphs apply therefore to any joint distribution generated in this way over a directed acyclic graph. A first path criterion was given by J. Pearl, who called it d-separation (for separation in directed graphs). Two other equivalent criteria are due to S. L. Lauritzen and coworkers, and D. R. Cox and N. Wermuth.